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The equation $F(D^2u) = g$ for $g \in C^{\alpha}$ should have a $C^{2,\alpha}$ estimate by perturbation theory. See for instance Caffarelli-Cabre, Ch. 8. The idea is that the constant-coefficient equa …

One approach is to observe that $$\|u_0\|_{L^{\infty}(B_1^+)} \geq \frac{1}{16n}|f(0)|.$$ It suffices by linearity to prove this when $f(0) = 1$. Consider the barrier $$b(x) = \frac{1}{2n}\left(\left| …

It is true and well-known (assuming $F$ is e.g. uniformly elliptic). The idea is sketched in ch. 9 of the book by Caffarelli-Cabre, but I am not sure of a precise reference at this level of generality …

Here is a counterexample: let $$u(x,y) = e^{-x^2}\sinh(y).$$ Then $$\det D^2u = -2e^{-2x^2}(\sinh^2(y) + 2x^2) < 0 \text{ ơn } \mathbb{R}^2 \backslash \{0\},$$ and the equation $$u_{xx} + (2-4x^2)u_{y …

A more general form of the problem is: $$\Delta u = g$$ with $g > 0$, $u \geq 0$ and $u(x',0) = 0$ for all $x' \in \mathbb{R}^{n-1}$. Here are some things we can say: 1) If $0 < g < K$ then by $W^{2, …

For constant coefficient linear equations, the question of regularity and existence is nicely answered by representation formulas such as the Poisson kernel, as you mentioned. These formulas tell us t …

It is possible to do this. Here is a sketch of the construction: 1) Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = …

A linear uniformly elliptic equation of the form $F(D^2u) = 0$ can only be $\Delta u = 0$ (up to an affine transformation of the solution) since the only inputs are the second derivatives. Equations l …

For the homogeneous equation, I have seen a proof of $C^{\alpha}$ regularity using an oscillation estimate based only on local boundedness and a Poincare-Sobolev inequality. Specifically: Let u be a …

The continuity of $\psi$ up to $\partial \Omega$ is false without more control on $R$. Consider for example the harmonic function that is $1$ on the upper half-circle and $-1$ on the lower half-circ …

In general the answer is "no." Let $\Omega = (-\pi/2,\,\pi/2) \subset \mathbb{R}$, and for $\varphi \in C^{\infty}_0(\Omega)$ take $$u_1 = \cos(x), \quad u_2 = \cos(x) + \epsilon \varphi$$ $$a_1 = 1, …

It depends on the geometry of the domain. Consider e.g. the function $f(x,\,y) = xy$ on $\{|xy| < 1\}$, which is bounded with constant Hessian on this domain but has linearly growing gradient. Or take …

The function $u(x) = e^{a \cdot x}$ is a positive solution for any vector $a$ in the sphere $\partial B_{1/2}\left(\frac{e_1}{2}\right)$. So is $u = e^{-\frac{x_1}{2}} w$ for any positive solution $w$ …

Not always. Consider the case $n \geq 5$, $K = B_1$, and $u = |x|^{\frac{4-n}{2}} - 1$. Then $u \in H^1_0(B_1)$ but $u \notin H^2(B_1)$, and $$\Delta u = \frac{n(4-n)}{4}(u+1)^{\frac{n}{n-4}} := c(u), …

$U = 2x^2-y^2$ and $u = x^2-2y^2$ solve constant-coefficient elliptic equations, but $U - u = x^2 + y^2$ has an interior minimum and thus cannot solve an elliptic equation.